Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2233 |
150 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(4,6,7,10)(5,9,8,11)(13,18)(15,23)(16,22)(17,25)(19,20)(21,27)(24,26);(1,2,3)(5,8)(9,11)(13,17)(18,25)(19,26)(20,24);(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,18)(16,24)(19,23)(20,22)(21,25)(26,27)Polycyclic presentation:
2^7,3|c^2=dg,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10708405, #10708406, #10708416, … |
| maximal normal subgroup of | #10708405, #10708406, #10708416, … |
| maximal quotient of | #10711477, #10711690, #10712126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3645);